Driving under the influence: the breathalyser fallacy

Abstract

The use of breathalysers to determine whether a person is ‘Driving Under the Influence’ (DUI) is an example of situations in which the outcome of a test – positive or negative – is used to determine whether a condition exists. As such, it is susceptible to a fallacy that is inherent in such situations: confusing the chances that a person would test positive if he/she was DUI with the chances that a person was DUI if he/she tested positive. This is referred to here as the breathalyser fallacy and this article provides quantitative estimates of the size of this fallacy. Almost all of the criticism of the conclusions of breathalyser tests (namely, a person over the limit is DUI, a person below the limit is not) concerns the likelihood of a person who is not DUI being incorrectly identified. This analysis questions this argument. We show that the likelihood of an innocent person testing positive depends on two further factors: (i) the a priori likelihood that a person selected for breath analysis is, in fact, DUI and (ii) the reliability of the test in terms of the likelihood of a DUI person being found to be over the limit. If these two likelihoods are high, then the probability of a person being guilty of DUI for a positive outcome on the breathalyser test would also be high, even though there is a significant chance that a person who is not DUI will be incorrectly identified as being over the limit.

Notes

¹An amalgam of ‘breath’ and ‘analysis’.

²More reliable methods are analysis of blood and/or urine samples with limits of 80 mg of alcohol in 100 mL of blood and 107 mg of alcohol in 100 mL of urine. However, because these tests are difficult to implement, the breathalyser remains the most common form of testing for DUI.

³A prosecutor argues that because the probability of observing a particular piece of evidence (say, blood type identical to that found at the scene of the crime), under the assumed innocence of the defendant, is very small the probability of the defendant being innocent, given that his blood type matches that at the crime scene, must also be very small. A doctor argues that because the probability of a person testing HIV positive, if he/she was HIV free, is very small, the probability of a patient being HIV free, given that he/she tested HIV positive, must also be very small. A labour market analyst argues that because only a small proportion of persons in regular employment are from a particular group, the probability of a person from that group being in regular employment must also be small.

⁴See ‘In Praise of Bayes’, The Economist, 28 September 2000.

⁵The updating factor is the ratio of the probability of observing the data when the theory is true to that of observing the data regardless of whether the theory is true or false: P(A) =  P(A|T)P(T)+P(A|)P(), () being the event that the theory is false.

⁶See for example, Buckley et al . (2001); Labianca (1990).

⁷For example, acetone, which dieters and diabetics have in much greater levels than others.